Much depends on your climate, and where you park the bike. As far south as I live, I have to bring all batteries inside for the summer, where it's not over 100F. It can get to 120 easy in the garage.
So all summer the chargers are in the house. But when it's cooler, I leave safe to charge batteries on the bike, but unplugged from the controller, and charge there in the garage. Once it gets really cold, it's back to keeping them inside, so they will be at 60F when I start riding. Too much sag and capacity loss leaving the house with a 30F battery.
I sometimes carry them in a bag, sometimes in a box. ALL batteries have a protective inner box, that I usually make by cutting and taping coroplast, that plastic cardboard they make political signs out of. This protects them from chafing and damage in the box or bag. Makes them more crash resistant.
One thing I never do, is build the battery permanently onto the bike. But my reason for this is I always have 4-6 ebikes ready to ride. So the battery hops from bike to bike constantly.